The two vectors u=⟨−1,−3,−1⟩ and v=⟨−2,3,1⟩ determine a plane in space. Which vectors lies in the same plane as u and v. A=⟨2,−3,1⟩ B=⟨0,3,0⟩ C=⟨6,−9,−3⟩ D=⟨−2,3,−3⟩

Answer 1

Hi Jessica,

Take the triple product of u and v with each of the other four vectors. For A, it looks like this: A dot (u cross v). For B, B dot (u cross v).

The cross product outputs a vector whose magnitude is the area of the parallelogram defined by the two vectors . Similarly, the triple product outputs a SCALAR that is the volume of the parallelepiped defined by the three vectors . A parallelepiped is like a rectangular prism but slanted, or more accurately, a rectangular prism is a parallelepiped that is straight up, with all 90 degree corners .

In any case, the triple product of three vectors is 0 if and only if the three vectors are coplanar. It makes sense that if you have three vectors that all lie in the same plane, they cannot define any solid that has volume . So whichever vectors give you 0 when you dot them with (u cross v) will be the one(s) that are coplanar with u and v . Hope this helps!

Robert